Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset’s elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation.

A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

The Out-of-Kilter Algorithm and Its Applications to Network Flow Problems

The out-of-kilter algorithm, which was published by D.R. Fulkerson [1], is an algorithm that computes the solution to the minimum-cost flow problem in a flow network. To begin, the algorithm starts with an initial flow along the arcs and a number for each of the nodes in the network. By making use of Complementary Slackness Optimality Conditions (CSOC) [2], the algorithm searches for out-of-kilter arcs (those that do not satisfy CSOC conditions). If none are found the algorithm is complete. For arcs that do not satisfy the CSOC theorem, the flow needs to be increased or decreased to bring them into kilter. The algorithm will look for a path that either increases or decreases the flow according to the need. This is done until all arcs are in-kilter, at which point the algorithm is complete. If no paths are found to improve the system then there is no feasible flow. The Out-of-Kilter algorithm is applied to find the optimal solution to any problem that involves network flows. This includes problems such as transportation, assignment and shortest path problems. Computer solutions using a Pascal program and Matlab are demonstrated.

A dual simplex method for grey linear programming problems based on duality results

PurposeFor extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.Design/methodology/approachThe authors discuss the solution concepts of primal and dual of grey linear programming problems without converting them to classical linear programming problems. A numerical example is provided to illustrate the theory developed.FindingsBy using arithmetic operations between interval grey numbers, the authors prove the complementary slackness theorem for grey linear programming problem and the associated dual problem.Originality/valueComplementary slackness theorem for grey linear programming is first presented and proven. After that, a dual simplex method in grey environment is introduced and then some useful concepts are presented.

An Efficient Algorithm for Solving Minimum Cost Flow Problem with Complementarity Slack Conditions

This paper presents an algorithm for solving a minimum cost flow (MCF) problem with a dual approach. The algorithm holds the complementary slackness at each iteration and finds an augmenting path by updating node potential iteratively. Then, flow can be augmented at the original network. In contrast to other popular algorithms, the presented algorithm does not find a residual network, nor find a shortest path. Furthermore, our algorithm holds information of node potential at each iteration, and we update node potential within finite iterations for expanding the admissible network. The validity of our algorithm is given. Numerical experiments show that our algorithm is an efficient algorithm for the MCF problem, especially for the network with a small interval of cost of per unit flow.

Optimal control of higher order differential inclusions with functional constraints

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Stochastic Bilevel Program for Optimal Coordinated Energy Trading of an EV Aggregator

Gradually replacing fossil-fueled vehicles in the transport sector with Electric Vehicles (EVs) may help ensure a sustainable future. With regard to the charging electric load of EVs, optimal scheduling of EV batteries, controlled by an aggregating agent, may provide flexibility and increase system efficiency. This work proposes a stochastic bilevel optimization problem based on the Stackelberg game to create price incentives that generate optimal trading plans for an EV aggregator in day-ahead, intra-day and real-time markets. The upper level represents the profit maximizer EV aggregator who participates in three sequential markets and is called a Stackelberg leader, while the second level represents the EV owner who aims at minimizing the EV charging cost, and who is called a Stackelberg follower. This formulation determines endogenously the profit-maximizing price levels constraint by cost-minimizing EV charging plans. To solve the proposed stochastic bilevel program, the second level is replaced by its optimality conditions. The strong duality theorem is deployed to substitute the complementary slackness condition. The final model is a stochastic convex problem which can be solved efficiently to determine the global optimality. Illustrative results are reported based on a small case with two vehicles. The numerical results rely on applying the proposed methodology to a large scale fleet of 100, 500, 1000 vehicles, which provides insights into the computational tractability of the current formulation.

Optimality of refraction strategies for a constrained dividend problem

We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.